Flattening Graphs

From:
andrew cooke <andrew@...>

Date:
Mon, 28 Mar 2011 06:46:16 -0300

When you construct a graph of relationships you often have a rather vague /
flexible notion of "relatedness". It's natural to use this to decide who are
neighbours, and even to weight the edges. But it also leads to a curious
problem when you start to use the graph.
The problem is that some nodes come out "better" than others. You can play
around with how you calculate your "relatedness factor", but the natural
result of any flexible measurement is a degree of inequality.
This can lead to bias. For example, in the case of Uykfe (and Uykfd before),
I construct a graph of related musicians and then use that to generate
playlists of related tracks. The "better" nodes tend to capture the playlist
(I imagine that you could relate this to eigenvectors of a transition matrix
or something similar?).
For example, Sparklehorse had links out to various artists, but the highest
weighted links from those neighbours went elsewhere. So my playlist, when it
starts at Sparklehorse, never returns.
My natural response to this is to make the graph undirected. But this doesn't
help - it turns out not of the links from Sparklehorse are that great, so
neighbours still prefer other nodes.
A simple solution (one I reinvented for Uykfe after using it in Uykfd and then
forgetting) is to restrict the number of links for each node to some fixed
amount (note: not a cutoff by weight, but a fixed number) and then ignore
weights in transitions.
This doesn't completely remove the problem - the "best" nodes now have more
connections that the "worst" - but it does level the playing field a little.
(I haven't yet tried weighting the notes out from nodes by the inverse of the
number of links at the target).
Andrew

Active Flattening

From:
andrew cooke <andrew@...>

Date:
Fri, 1 Apr 2011 08:58:02 -0300

Just to follow up on this, the latest (final?) version of Uykfe does actively
flatten graphs: it initially generates 10 outgoing edges per node. Then it
identifies nodes with the most incoming edges and deletes the loweest weighted
edges that contribute to that. It's not a perfect solution (you have to stop
when you hit some minimum number of outgoing edges in the neighbours), but it
helps.
Andrew